The present invention relates to a method of using Differential Scanning Calorimetry (xe2x80x9cDSCxe2x80x9d) to measure the thermal conductivity of materials.
Thermal conductivity characterizes the ability of a material to conduct heat. Traditional methods for measuring the thermal conductivity of materials comprise imposing a temperature gradient upon a material of known geometry, and measuring the heat flow through the material. The heat flow is measured by, for example, measuring the temperature drop across a sheet of material having a known thermal conductivity. Traditional methods of measuring thermal conductivity are limited in that they assume that thermal contact of a sample is highly reproducible and determinable from a calibration measurement. The present invention overcomes this limitation by using a simultaneous thermal contact measurement. No method has been heretofore proposed to determine the actual thermal contact and the thermal conductivity simultaneously from a single measurement of a heat capacity spectrum.
Thermal conductivity data are in great demand by industry, for use in polymer injection molding, in encapsulation of electronic devices and, in general, in modeling of different processes. Nowadays commercial techniques often measure thermal diffusivity or effusivity and calculate thermal conductivity using heat capacity values, measured separately.
DSC is a commercially available and widely used technique to measure heat capacity of samples of milligram size in a wide temperature range. Therefore it would be opportune to add to DSC instruments a feature to measure thermal conductivity of typical DSC samples.
Prior art of interest includes P. G. Knibbe, J. Phys. E: Sci. Instrum., vol. 20, pp. 1205-1211 (1987) which describes a xe2x80x9chot wirexe2x80x9d technique for measuring the thermal conductivity of a material. This technique uses a temperature-sensitive resistor wire embedded in a sample of the material. The resistor wire serves the dual function of supplying heat to the specimen, and measuring the temperature change at the wire. This rate of change is related to the thermal conductivity of the sample of the material.
D. G. Cahill and R. O. Pohl, Phys. Rev. B. vol. 35, p. 4067 (1987), and D. G. Cahill, Rev. Sci. Instrum. vol. 61(2), pp. 802-808 (1990), describe a xe2x80x9c3xcfx89xe2x80x9d technique for measuring thermal conductivity. This technique uses a temperature sensitive resistive metal film evaporated as a narrow line onto the surface of the sample to simultaneously heat the sample and detect the flow of heat away from the metal line. A current at angular frequency xcfx89 heats the metal line at a frequency of 2 xcfx89. Because the resistance of a metal increases with increasing temperature, and this temperature is modulated by the sample thermal conductivity, this produces a small oscillation in the resistance of the metal line, resulting in a voltage across the resistor at a frequency of 3 xcfx89. The thermal conductivity of the sample is then calculated from the amplitude of the 3xcfx89 voltage oscillations.
J. H. Flynn and D. M. Levin, Thermochimica Acta, vol. 126, pp. 93-100 (1988), describes a thermal conductivity measurement method, suitable for measuring the thermal conductivity of sheet materials, based upon first-order transitions in a sensor material. A film of the sensor material is placed on a surface of the sheet material. The thermal conductivity measurement is made at the temperature at which the sensor material undergoes a first order transition. For example, if indium is used as the sensor material, the measurement is made at the melting point of indium, i.e., at the temperature at which indium undergoes a first order transition. The flow of heat into the sensor material must match the transition enthalpy. The thermal conductivity of the sheet material is obtained by comparing the data obtained with only the sensor material in the heater of a differential scanning calorimeter, to the data obtained with the sensor material on top of the sheet material in the differential scanning calorimeter.
T. Hashimoto, Y. Matsui, A. Hagiwara and A. Miyamoto; Termochimica Acta vol. 163, pp. 317-324 (1990), describes a method to obtain thermal diffusivity by an AC calorimetric method. The AC current is passed through the heater; the periodical heat flow generates and diffuses to the rear surface of the sample. The variation of the temperature at the rear surface was detected. A sputtered gold layer on both surfaces was used as the heater and the sensor of temperature variation. The thermal diffusivity of four polymers were measured over the temperature range 20-200xc2x0 C.
S. M. Marcus and R. L. Blaine, Thermochimica Acta, vol. 243, pp. 231-239 (1994) (herein incorporated by reference), describes a thermal conductivity measurement method where thermal conductivity is measured without modification of the commercially available DSC cell. A calculation of thermal conductivity is determined from a ratio of apparent and true heat capacities measured from a thick (about 3 mm) and a thin (about 0.5 mm) sample, respectively, in temperature-modulated mode. An additional calibration step takes heat losses through the purge gas surrounding into account. This method is based on the assumption that the face of the specimen at the heat source follows the applied temperature modulation, which means no thermal resistance between the sample and the furnace.
S. L. Simon and G. B. McKenna, J. Reinforced Plastics Composites, vol. 18, pp. 559-571 (1999) (herein incorporated by reference), describes two problems in the aforesaid method of Marcus and Blaine. First, the equation relating the apparent heat capacity to the thermal conductivity is limited in range due to an approximation made in their derivation. Second, a thermal resistance between the sample and the furnace can have significant effect on the measured apparent heat capacity. This reference also teaches that when calculating a value for thermal conductivity it is necessary to know the heat transfer coefficient, for without it, an accurate value cannot be obtained.
S. L. Simon and G. B. McKenna, J. Reinforced Plastics Composites, vol. 18, pp. 559-571 (1999), as well as U.S. Pat. No. 5,335,993 to Marcus et al. (herein incorporated by reference), additionally describe a method of determining thermal conductivity by obtaining the value from a single run at several frequencies, i.e. it is not necessary to measure two samples. In U.S. Pat. No. 5,335,993 the method described therein has insufficient sensitivity, in part; because it measures the conductivity of massive bodies brought into thermal contact with thin film wafers and does not fully eliminate interface thermal resistance. The basic problem with this patent as well as S. L. Simon and G. B. McKenna, J. Reinforced Plastics Composites, vol. 18, pp. 559-571 (1999), is that the actual thermal contact between sample and oven is not considered in the calculation of thermal conductivity. Theses references teach that thermal contact is considered to be highly reproducible and determined from the calibration measurement.
U.S. Pat. No. 5,244,775 to Reading et al. and U.S. Pat. No. 5,439,291 to Reading utilize frequency measurement resulting from the application of heat to determine phase transition of materials.
U.S. Pat. No. 5,439,291 to Reading (herein incorporated by reference) describes a technique for determining physical properties of a sample using thermal modulation techniques. Two identical samples are used, with one experiencing a linear temperature ramp and the other experiencing the same ramp with a temperature oscillation imposed. A chopped light source can be used to provide the energy necessary for the temperature oscillation. Thermocouples attached to each sample measure the temperature of each sample. Light is used as a radiation source to heat the temperature-modulated sample.
Of interest is Pat. No. 5,688,049 to Gorvorkov, which teaches a device and method for measuring the thermal conductivity of a thin film by determining the change in temperature near the surface of the film after a sample including the film is illuminated with a beam of light. This method is accomplished by modulating the beam of light at a selected modulation frequency and measuring the amplitude of the sound waves created in the gas near the surface of the sample as a result of the repetitive heating and cooling of the surface.
These techniques are all subject to significant limitations. For example, the hot wire technique requires large samples, long times for the sample to come into equilibrium, and an additional long measurement period. The AC and the 3xcfx89 techniques require a thin metal film in intimate contact with the sample, with fine electrical contacts to the film. The metal film must be thermally isolated from any heat sink, except for the sample being measured. The combination of film and sample is not mechanically robust, and is not readily separable so that other samples can be measured. The first order transition technique is restricted to the temperatures where materials are available with sharp first order transitions.
Accordingly, it is desirable to provide a simple, rapid and nondestructive technique for measuring the thermal conductivity of low thermal conducting solid materials, particularly one that is sensitive enough to measure the conductivity of materials with K in the range of 0.1-2 W mxe2x88x921 Kxe2x88x921, and is suitable for on-line control in an industrial production setting and overcomes other drawbacks of the prior art. Since thermal contact cannot be reproduced an improved method to determine the actual thermal contact and the thermal conductivity simultaneously from the heat capacity spectrum from a single measurement is disclosed. No thermal conductivity calibrant is necessary.
The present method determines effective heat capacity Ceff(xcfx89) at different frequencies calculated using a novel step response analysis as a ratio of heat flow rate amplitude AHF and heating rate amplitude Aq. With this model one can thoroughly describe the dynamic behavior of DSC and temperature modulated DSC (TMDSC) systems under conditions of an appreciable temperature gradient inside the sample. A novel algorithm to determine the most important parameters: sample heat capacity, effective thermal contact between the sample and the furnace and finally sample thermal conductivity for the case of real valued heat capacity and thermal conductivity is disclosed.
The present invention provides for quick measurements of small samples for thermal conductivity and heat capacity of the sample are determined simultaneously in a single measurement with the prerequisite that these values are frequency independent. The method is appropriate for state-of-the art power-compensated differential scanning calorimeters without any modification of the measuring system.
xe2x80x9cThermal conductivity,xe2x80x9d as used herein, is the ratio of the heat flow per unit area in the sample to the temperature gradient in the sample.
xe2x80x9cSpecific heat,xe2x80x9d as used herein, is the ratio of the change in heat content of a sample of uniform temperature to the product of the change in temperature of the sample and mass of the sample.
xe2x80x9cEffective heat capacity,xe2x80x9d as used herein, is the ratio of the amplitude of the heat flow into a sample to the amplitude of the heating rate applied to the sample at the heat source.
xe2x80x9cThermal contact,xe2x80x9d as used herein, is the ratio of the heat flow through the sample bottom surface to the temperature difference between furnace and sample.
xe2x80x9cStep response analysisxe2x80x9d, as used herein, is the measurement of heat flow rate and heating rate in time domain.
The present invention uses known DSC equipment to provide novel techniques for measuring the thermal conductivity of low thermal conducting solid materials, particularly one that is sensitive enough to measure the conductivity of materials with thermal conductivity in the range of about 0.1 W mxe2x88x921 Kxe2x88x921 to about 2 W mxe2x88x921 Kxe2x88x921.
A system exists in a furnace system where:
It is the object of the present invention to teach a method to determine simultaneously thermal conductivity, the thermal contact and heat capacity of low thermal conducting materials like polymers. No additional measurements, no thermal conducting calibrants, and no modifications of the commercially available DSC are necessary.
It is the object of the present invention to provide a method based on temperature waves propagation through the sample. At low frequencies of perturbation temperature waves go through whole sample without damping, the whole sample is modulated and, therefore, large effective heat capacity is measured. At higher frequencies, temperature waves are damped and the sample is modulated only partly. Thus, measured effective heat capacity is smaller. The damping is stronger for poor thermal conducting materials.
It is the object of the present invention to provide a method, which overcomes the disadvantage of known techniques for measuring thermal contact.
It is the object of the present invention to provide a method, which determines the actual thermal contact and the thermal conductivity simultaneously from a heat capacity spectrum from a single measurement. No thermal conductivity calibrant is necessary.
It is the object of the present invention to provide a method for measuring thermal conductivity where heat capacity of the sample is measured at different frequencies. Low frequency temperature waves go through the whole sample without damping, the whole sample is modulated and therefore, a large apparent heat capacity is measured. Higher frequency temperature waves are damped and the sample is partly modulatedxe2x80x94the measured apparent heat capacity is smaller. The damping is stronger for poor thermal conducting materials. However, the same damping effect appears by finite thermal contact (heat transfer coefficient) between the sample surface and the furnacexe2x80x94a poor thermal contact damps the temperature waves more strongly than a good one. But the thermal contact and thermal conductivity lead to different frequency dependencies of apparent heat capacity. This difference allows the necessary separating of damping effects due to the thermal contact and due to the thermal conductivity. Moreover, it is not necessary to measure apparent heat capacity at different frequencies with TMDSC. The spectrum of temperature waves can be easily generated by a single step in the program temperature.
In the preferred embodiment of this invention, known DSC apparatuses are utilized, and may be simplified to cases where a solid sample is measured directly in a DSC furnace. Three parameters need to be determined: specific heat capacity, effective thermal contact between the sample and the furnace (K), and thermal conductivity (xcexa).
It is the object of the present invention to provide a method of measuring thermal conductivity of a sample where effective heat capacity Ceff(xcfx89) at different frequencies can be calculated from step response analysis as a ratio of heat flow rate amplitude AHF and heating rate amplitude Aq:                                           C            eff                    ⁡                      (            ω            )                          =                                            A                              H                ⁢                                  xe2x80x83                                ⁢                F                                                    A              q                                =                                                                      ∑                                      i                    =                    1                                    n                                ⁢                                  H                  ⁢                                      xe2x80x83                                    ⁢                                      F                    i                                    ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          ω                      ⁢                                              xe2x80x83                                            ⁢                                              t                        i                                                              )                                                              -                              i                ⁢                                                      ∑                                          i                      =                      1                                        n                                    ⁢                                      H                    ⁢                                          xe2x80x83                                        ⁢                                          F                      i                                        ⁢                                          sin                      ⁡                                              (                                                  ω                          ⁢                                                      xe2x80x83                                                    ⁢                                                      t                            i                                                                          )                                                                                                                                                                  ∑                                      i                    =                    1                                    n                                ⁢                                                      q                    i                                    ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          ω                      ⁢                                              xe2x80x83                                            ⁢                                              t                        i                                                              )                                                              -                              i                ⁢                                                      ∑                                          i                      =                      1                                        n                                    ⁢                                                            q                      i                                        ⁢                    sin                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              ω                        ⁢                                                  xe2x80x83                                                ⁢                                                  t                          i                                                                    )                                                                                                                              (        1        )            
where points of heat flow rate, HFi, and heating rate, qi, should be taken both with the same sampling rate (number of points per unit time). Points are collected from the beginning of the temperature step until the heat flow reaches the steady state value at the isotherm, having in total n points. After that the Ceff(xcfx89) values are corrected for apparatus influence (instrumental delay) as:
Capp(xcfx89)=Ceff(xcfx89)xc2x7B2(xcfx89)xe2x80x83xe2x80x83(2) 
where Capp(xcfx89) is an apparent heat capacity at frequency xcfx89, and B2(xcfx89) is the dynamic calibration factor of the instrument.
It is the object of the present invention to provide a method of measuring thermal conductivity where the first parameter of the system, the specific heat capacity cp, can easily be determined as                               c          p                =                                            C              eff                        ⁡                          (              0              )                                            m            s                                              (        3        )            
where ms is the sample mass and Ceff(0) is calculated from Eg. (1) for xcfx89=0.
It is the object of the present invention to provide a method of measuring thermal conductivity where apparent heat capacity is given as                                           C            app                    ⁡                      (            ω            )                          =                                            C              α                        ⁡                          (              ω              )                                            1            -                                                            i                  ⁢                                      xe2x80x83                                    ⁢                  ω                                K                            ⁢                                                C                  α                                ⁡                                  (                  ω                  )                                                                                        (        4        )            
where Cxcex1(xcfx89) is the apparent heat capacity in a case of ideal thermal contact between the sample and the furnace. Apparent heat capacity is measured directly on the bottom surface of the sample.
Equation (4) from above can be re-written as:                                           C            α                    ⁡                      (            ω            )                          =                                            C              app                        ⁡                          (              ω              )                                            1            +                                                            i                  ⁢                                      xe2x80x83                                    ⁢                  ω                                K                            ⁢                                                C                  app                                ⁡                                  (                  ω                  )                                                                                        (        5        )            
Unknown parameter in Eq. (5) is K because Capp(xcfx89) is measured by DSC. The lower the frequency xcfx89k the larger the modulus of Capp(xcfx89k) and Cxcex1(xcfx89k).
It is the object of the present invention to provide a method of measuring thermal conductivity where the second parameter (effective thermal contact of said sample) of the system is determined by describing Cxcex1(xcfx89) on a solid curve. The theoretical Cxcex1(xcfx89) curve in a polar plot representation depends only on the value Cxcex1(xcfx89=0), that is sample true heat capacity cp*ms, and does not depend on all other parameters. The correct value for K is then the value at which all Cxcex1(xcfx89k) points, calculated by Eq. (5), lie on the theoretical curve.
It is the object of the present invention to provide a method of measuring thermal conductivity where sample thermal conductivity xcexa is readily determined.                                           C            α                    ⁡                      (            ω            )                          =                              -                          1                              i                ⁢                                  xe2x80x83                                ⁢                ω                                              ⁢                      κ            ·            S            ·            α                    ⁢                      xe2x80x83                    ⁢                      tanh            ⁡                          (                              α                ·                d                            )                                                          (        6        )            
where all measured parameters, except thermal conductivity xcexa, are known (density xcfx81 can be calculated from the sample mass and sample size, which are set before measurement). At any given frequency xcfx89kxe2x89xa00 increasing of xcexa leads to shifting the position of the Cxcex1(xcfx89k) point on the theoretical curve towards Cxcex1(xcfx89=0).
It is the object of the present invention to provide a method of measuring thermal conductivity where by varying xcexa in                                           C            α                    ⁡                      (            ω            )                          =                              -                          1                              i                ⁢                                  xe2x80x83                                ⁢                ω                                              ⁢                      κ            ·            S            ·            α                    ⁢                      xe2x80x83                    ⁢                      tanh            ⁡                          (                              α                ·                d                            )                                                          (        6        )            
the condition is reached where the same set of xcfx89k calculated Cxcex1(xcfx89k) points coincide with measured points Cxcex1(xcfx89k), determined by Eq. (5).
The measured points for Cxcex1(xcfx89k) will not exactly coincide with theoretical ones, rather a scatter corresponding to only about 1 to about 2 percent uncertainties in determination of thermal conductivity (xcexa) and effective thermal contact (K) is produced.